Large deviation estimates and H\"older regularity of the Lyapunov exponents for quasi-periodic Schr\"odinger cocycles

TL;DR

This paper establishes large deviation estimates and proves optimal H"older regularity of Lyapunov exponents for quasi-periodic Schr"odinger operators with analytic potentials, covering various frequency regimes.

Contribution

It provides the first comprehensive proof of large deviation estimates and H"older continuity of Lyapunov exponents for a broad class of quasi-periodic Schr"odinger operators.

Findings

Large deviation estimates in positive Lyapunov exponent regime

H"older continuity of Lyapunov exponents and integrated density of states

Results valid for all Diophantine and some Liouville frequencies

Abstract

We consider one-dimensional quasi-periodic Schr\"odinger operators with analytic potentials. In the positive Lyapunov exponent regime, we prove large deviation estimates which lead to optimal H\"older continuity of the Lyapunov exponents and the integrated density of states, in both small Lyapunov exponent and large coupling regimes. Our results cover all the Diophantine frequencies and some Liouville frequencies.